The posterior covariance \(\hat{V}_n(\mathbf {x},\mathbf {x}^{\prime })\)  can be viewed as the bias of a noise-free KRR estimator [1]}. To see this, let \(\hat{K}_{\mathbf {x}^{\prime }}\)  be the KRR estimator with noiseless observations of \(K_{\mathbf {x}^{\prime }}\)  at \(\lbrace X_i\rbrace _{i=1}^n\) , i.e.,
\(\hat{K}_{\mathbf {x}^{\prime }} = \underset{f \in \mathbb {H}}{\arg \min } \left\lbrace \frac{1}{n}\sum _{i = 1}^n (K(X_i,\mathbf {x}^{\prime }) - f(X_i))^2 + \lambda \Vert f\Vert ^2_{\mathbb {H}}\right\rbrace .\) 
